Stochastic sewing with Besov regularity

Under various conditions, sewing lemmas provide convergence of the Riemann-type sum Σ[s,t]∈π Ξs,t for a given two-parametric map Ξ as the mesh sizes of the considered partitions π tend to zero. In this note, we prove a stochastic sewing lemma for two-parameter processes whose increments, when viewed as functions with values in Lm(Ω;𝕍) for m ≥ 2 and a real separable Banach space 𝕍 with a non-trivial martingale type, are of Besov regularity. The contribution is two-fold: First, we generalize the stochastic sewing lemma of Lê [Electron. J. Probab. 25 (2020) 1–55] for processes whose increments belong to a Besov and not necessarily Hölder space. Second, we show here that the assumptions of the Besov sewing lemma of Friz et al. [J. Differ. Equ. 339 (2022) 152–231] can be relaxed if stochastics is incorporated in the sewing from the beginning. As an application, Besov regularity of the Itˆo integral of Brownian functionals is obtained.